A302491 -id:A302491 - cp's OEIS frontend

A302591 One, powers of 2, and prime numbers of squarefree index.

1, 2, 3, 4, 5, 8, 11, 13, 16, 17, 29, 31, 32, 41, 43, 47, 59, 64, 67, 73, 79, 83, 101, 109, 113, 127, 128, 137, 139, 149, 157, 163, 167, 179, 181, 191, 199, 211, 233, 241, 256, 257, 269, 271, 277, 283, 293, 313, 317, 331, 347, 349, 353, 367, 373, 389, 397, 401

Offset: 1

Gus Wiseman, Apr 10 2018

nonn

A prime index of n is a number m such that prime(m) divides n.

			Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of set systems.
01: {}
02: {{}}
03: {{1}}
04: {{},{}}
05: {{2}}
08: {{},{},{}}
11: {{3}}
13: {{1,2}}
16: {{},{},{},{}}
17: {{4}}
29: {{1,3}}
31: {{5}}
32: {{},{},{},{},{}}
41: {{6}}
43: {{1,4}}
47: {{2,3}}
59: {{7}}
64: {{},{},{},{},{},{}}

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000079, A000961, A001222, A003963, A005117, A007716, A056239, A275024, A279791, A281113, A296133, A301765, A302242, A302243, A302491.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[400],Or[#===1,Union[primeMS[#]]==={1},PrimeQ[#]&&SquareFreeQ[PrimePi[#]]]&]

ok(n)={n>>valuation(n,2) == 1 || (isprime(n) && issquarefree(primepi(n)))} \\ Andrew Howroyd, Aug 26 2018

Union of A000079 and A302491. - Andrew Howroyd, Aug 26 2018

A302979 Powers of squarefree numbers whose prime indices are relatively prime. Heinz numbers of uniform partitions with relatively prime parts.

Original entry on oeis.org

2, 4, 6, 8, 10, 14, 15, 16, 22, 26, 30, 32, 33, 34, 35, 36, 38, 42, 46, 51, 55, 58, 62, 64, 66, 69, 70, 74, 77, 78, 82, 85, 86, 93, 94, 95, 100, 102, 105, 106, 110, 114, 118, 119, 122, 123, 128, 130, 134, 138, 141, 142, 143, 145, 146, 154, 155, 158, 161, 165

Offset: 1

Gus Wiseman, Apr 16 2018

nonn

A prime index of n is a number m such that prime(m) divides n. An integer partition is uniform if all parts appear with the same multiplicity. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The number of uniform partitions of n with relatively prime parts is A078374(n).

			Sequence of all uniform relatively prime integer partitions begins (1), (11), (21), (111), (31), (41), (32), (1111), (51), (61), (321), (11111), (52), (71), (43), (2211).

A. David Christopher and M. Davamani Christober, Relatively Prime Uniform Partitions, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp.1-12.

Cf. A000009, A000837, A007916, A047966, A052409, A052410, A072774, A078374, A289023, A289509, A300486, A302491, A302796.

Select[Range[200],And[GCD@@PrimePi/@FactorInteger[#][[All,1]]===1,SameQ@@FactorInteger[#][[All,2]]]&]

A368603 Products of odd primes of squarefree index. MM-numbers of set multipartitions.

Original entry on oeis.org

1, 3, 5, 9, 11, 13, 15, 17, 25, 27, 29, 31, 33, 39, 41, 43, 45, 47, 51, 55, 59, 65, 67, 73, 75, 79, 81, 83, 85, 87, 93, 99, 101, 109, 113, 117, 121, 123, 125, 127, 129, 135, 137, 139, 141, 143, 145, 149, 153, 155, 157, 163, 165, 167, 169, 177, 179, 181, 187

Offset: 1

Gus Wiseman, Jan 08 2024

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

A set multipartition is a finite multiset of finite nonempty sets.

			The terms together with the corresponding set multipartitions begin:
   1: {}
   3: {{1}}
   5: {{2}}
   9: {{1},{1}}
  11: {{3}}
  13: {{1,2}}
  15: {{1},{2}}
  17: {{4}}
  25: {{2},{2}}
  27: {{1},{1},{1}}
  29: {{1,3}}
  31: {{5}}
  33: {{1},{3}}
  39: {{1},{1,2}}
  41: {{6}}
  43: {{1,4}}
  45: {{1},{1},{2}}

Odd case of A302478.
Products of odd terms of A302491.
A049311 counts non-isomorphic set multipartitions, strict A283877.
A050320 counts set multipartitions of prime indices.
A056239 adds up prime indices, row sums of A112798.
A089259 counts set multipartitions of integer partitions.
A116540 counts set multipartitions covering an initial interval by weight.
A368533 lists numbers with squarefree binary indices.
Cf. A000040, A000720, A001222, A005117, A006450, A076610, A270995, A296119, A302242, A302590, A339113.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],OddQ[#]&&And@@SquareFreeQ/@prix[#]&]

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A302591 One, powers of 2, and prime numbers of squarefree index.

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A302979 Powers of squarefree numbers whose prime indices are relatively prime. Heinz numbers of uniform partitions with relatively prime parts.

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A368603 Products of odd primes of squarefree index. MM-numbers of set multipartitions.

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Author

Keywords

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Examples

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Programs

Mathematica